回復 8# 阿光 的帖子
第10題
\(\begin{align}
& y={{x}^{3}}-2{{x}^{2}},y'=3{{x}^{2}}-4x \\
& y=4{{x}^{2}}-k,y'=8x \\
\end{align}\)
令\(P\left( t,{{t}^{3}}-2{{t}^{2}} \right),k=4{{t}^{2}}-\left( {{t}^{3}}-2{{t}^{2}} \right)=-{{t}^{3}}+6{{t}^{2}}>0\)
過\(P\)之公切線斜率
\(\begin{align}
& =3{{t}^{2}}-4t=8t \\
& t=4 \\
& k=32 \\
\end{align}\)
\(y={{x}^{3}}-2{{x}^{2}}\)和\(y=4{{x}^{2}}-32\)交於\(\left( -2,-16 \right),\left( 4,32 \right)\)
所求\(=\int_{-2}^{4}{\left[ {{x}^{3}}-2{{x}^{2}}-\left( 4{{x}^{2}}-32 \right) \right]}dx=108\)
